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The goal of the PLL is to maintain a demodulating sine and cosine that match the incoming carrier. Suppose ${\omega}_{c}$ is the believed digital carrier frequency. We can then represent the actual received carrier frequency as theexpected carrier frequency with some offset, $\tilde{{\omega}_{c}}={\omega}_{c}+\tilde{\theta}(n)$ . The NCO generates the demodulating sine and cosine with the expected digital frequency ${\omega}_{c}$ and offsets this frequency with the output of the loop filter. The NCO frequency can then be modeled as $\hat{{\omega}_{c}}={\omega}_{c}+\hat{\theta}(n)$ . Using the appropriate trigonometric identities $\cos A\cos B=1/2(\cos (A-B)+\cos (A+B))$ and $\cos A\sin B=1/2(\sin (B-A)+\sin (A+B))$ . , the in-phase and quadrature signals can be expressed as
The estimated phase mismatch estimate is fed to the NCO via a loop filter, often a simple low-pass filter. For thisexercise you can use a one-tap IIR filter,
It is suggested that you start by choosing $\alpha =0.6$ and $K=0.15$ for the loop gain. Once you have a working system, investigate the effects of modifying these values.
Simulate the PLL system shown in [link] using MATLAB. As with the DLL simulation, you will have to simulate the PLL on a sample-by-sample basis.
Use [link] to implement your NCO in MATLAB. However, to ensure that the phase term does not grow toinfinity, you should use addition modulo $2\pi $ in the phase update relation. This can be done by setting $\theta (n)=\theta (n)-2\pi $ whenever $\theta (n)> 2\pi $ .
[link] illustrates how the proposed PLL will behave when given a modulated BPSK waveform. In this case thetransmitted carrier frequency was set to $\tilde{{\omega}_{c}}=\frac{\pi}{2}+\frac{\pi}{1024}$ to simulate a frequency offset.
Note that an amplitude transition in the BPSK waveform is equivalent to a phase shift of the carrier by $\frac{\pi}{2}$ . Immediately after this phase change occurs, the PLL begins to adjust the phase to force the quadraturecomponent to zero (and the in-phase component to $1/2$ ). Why would this phase detector not work in a real BPSK environment? How could it be changed to work?
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